Linear Equations and Inequalities

Linear Equations

 

Just like two numbers in an ordered pair can represent a point, a line can be represented numerically as well.  A line is represented by an equation, which has variables x and y in it.  Any ordered pair that sits on that line will have an x and y coordinate that work when plugged into the equation.  Since lines continue on forever, the number of possible points that could fit in an equation is endless.

 

If you were given a point and an equation and were asked if the point is on that line, you could find the answer by plugging in the x and y coordinate of the point into the equation.

 

Is the point (3, 5) on the line y = 2x – 1?

 

Plug in the values for x and y:

5 = 2(3) – 1

5 = 6 – 1

5 = 5

Yes, (3, 5) is on the line y = 2x – 1.

 

T-Charts

 

One way to draw a line from an equation is to make a list of points on the line by creating a t-chart.  In an x-y t-chart, make up values for x, plug them into the equation, and find the missing values for y.  Then, plot those points and draw the line. 

 

Draw the line for the equation y = 2x – 1

 

Make up some values for x.  You can use any numbers, but smaller numbers will usually be easier

x

y

0

 

1

 

2

 

 

Plug in each of those number in the equation and solve for y:

y = 2(0) – 1

y = 0 – 1

y = -1

 

y = 2(1) – 1

y = 2 – 1

y = 1

 

y = 2(2) – 1

y = 4 – 1

y = 3

 

Now we see what y is for each of the x values we chose.  We can fill in our chart:

x

y

0

-1

1

1

2

3

 

Now, we plot these ordered pairs:

 

 

Slope-Intercept Form

 

While the t-chart method is necessary sometimes, there are formulas we can use to gather information from an equation and plot a line.  One of these is slope-intercept form, which is on your formula sheet.  The formula says that an equation is in the form y = mx + b, where m is the slope and b is the y-intercept.  The y-intercept is the point at which the line crosses the y-axis.  In order to use slope-intercept form to draw a line, you must set the equation equal to y, or get y alone on one side of the equals sign and combine all like terms.

 

Plot the line y = 2x – 3

 

First locate the y-intercept.  It is the number that is being added to the term with the x in it (or the number in the ‘b’ position in the formula.  In this equation, the y-intercept is -3.  You can plot that point on the y-axis as (0,-3).  Then use the slope given in the formula to plot more points.  The slope (in the ‘m’ position) of this equation is 2, or , which means you will rise 2 and run 1.  Your next point is then (-1, 1).  You can plot more points, or begin drawing your line.

 

 

You can also use slope intercept form to write the equation from a line.  If you were only given the picture shown, you could identify that the y-intercept is -3 by seeing the point where the line crosses the y-axis.  You could also identify the slope by seeing that the line rises 2 blocks for every 1 that it runs, so the slope is 2.  Then write the equation using the formula y = mx + b and substituting the y-intercept for b and the slope for m:  y = 2x – 3.

 

Point-Slope Form

 

What if you were only given the slope of a line and one point on the line, but the point is not the y-intercept.  In this case you could write the equation using a formula called point-slope form, which is also on your formula sheet.  The formula is  y1 = m( x1)y1 and x1 are where you plug in the coordinates of the point you are given.  m is the slope.

 

 

Linear Inequalities

 

Linear inequalities are drawn the same way as linear equations, but since they include not only points exactly on the line, but may include points greater than or less than the line, part of the graph is shaded. 

 

If the inequality is greater than (y > x), a dashed line is used (since points on the line are not included) and the area above the line is shaded.

 

If the inequality is less than (y < x), a dashed line is used and the area below the line is shaded.

If the inequality is greater than or equal to (y  x), a solid line is used and the area above the line is shaded.

If the inequality is less than or equal to (y  x), a solid line is used and the area below the line is shaded.